Journal of Topology最新文献

Given a lattice Veech group in the mapping class group of a closed surface $�S$, this paper investigates the geometry of $�\Gamma$, the associated $��{\pi }_1�S�$-extension group. We prove that $�\Gamma$ is the fundamental group of a bundle with a singular Euclidean-by-hyperbolic geometry. Our main result is that collapsing “obvious” product regions of the universal cover produces an action of $�\Gamma$ on a hyperbolic space, retaining most of the geometry of $�\Gamma$. This action is a key ingredient in the sequel where we show that $�\Gamma$ is hierarchically hyperbolic and quasi-isometrically rigid.

We provide a sufficient condition for splitness of a link in terms of its reduced $��\mathrm{sl}�\left(�N�\right)�$ link homology with arbitrary field coefficients. The proof of sufficiency uses Dowlin's spectral sequence and sutured Floer homology with twisted coefficients. If $�N$ is prime and the coefficient field is of characteristic $�N$, then the sufficient condition for splitness is also necessary. When $��N�=�2�$, we recover Lipshitz–Sarkar's split link detection result for Khovanov homology with $��Z�/�2�$ coefficients.

Landry, Minsky and Taylor defined the taut polynomial of a veering triangulation. Its specialisations generalise the Teichmüller polynomial of a fibred face of the Thurston norm ball. We prove that the taut polynomial of a veering triangulation is equal to a certain twisted Alexander polynomial of the underlying manifold. Thus, the Teichmüller polynomials are just specialisations of twisted Alexander polynomials. We also give formulae relating the taut polynomial and the untwisted Alexander polynomial. There are two formulae, depending on whether the maximal free abelian cover of a veering triangulation is edge-orientable or not. Furthermore, we consider 3-manifolds obtained by Dehn filling a veering triangulation. In this case, we give formulae that relate the specialisation of the taut polynomial under a Dehn filling and the Alexander polynomial of the Dehn-filled manifold. This extends a theorem of McMullen connecting the Teichmüller polynomial and the Alexander polynomial to the non-fibred setting, and improves it in the fibred case. We also prove a sufficient and necessary condition for the existence of an orientable fibred class in the cone over a fibred face of the Thurston norm ball.

We give an obstruction to positive scalar curvature metrics on 4-manifolds with the homology $��S^1�\times �S^3�$ described in terms of homology cobordism invariants from Seiberg–Witten theory. The main tool of the proof is a relative Bauer–Furuta-type invariant on a periodic-end 4-manifold.

We define a relative version of the Turaev–Viro invariants for an ideally triangulated compact 3-manifold with nonempty boundary and a coloring on the edges, generalizing the Turaev–Viro invariants [36] of the manifold. We also propose the volume conjecture for these invariants whose asymptotic behavior is related to the volume of the manifold in the hyperbolic polyhedral metric [22, 23] with singular locus of the edges and cone angles determined by the coloring, and prove the conjecture in the case that the cone angles are sufficiently small. This suggests an approach of solving the volume conjecture for the Turaev–Viro invariants proposed by Chen–Yang [8] for hyperbolic 3-manifolds with totally geodesic boundary.

We show that the sheaf of $�A^1$-connected components of a reductive algebraic group over a perfect field is strongly $�A^1$-invariant. As a consequence, torsors under such groups give rise to $�A^1$-fiber sequences. We also show that sections of $�A^1$-connected components of anisotropic, semisimple, simply connected algebraic groups over an arbitrary field agree with their $�R$-equivalence classes, thereby removing the perfectness assumption in the previously known results about the characterization of isotropy in terms of affine homotopy invariance of Nisnevich locally trivial torsors.

We show that for a closed hyperbolic 3-manifold, the size of the first eigenvalue of the Hodge Laplacian acting on coexact 1-forms is comparable to an isoperimetric ratio relating geodesic length and stable commutator length with comparison constants that depend polynomially on the volume and on a lower bound on injectivity radius, refining estimates of Lipnowski and Stern. We use this estimate to show that there exist sequences of closed hyperbolic 3-manifolds with injectivity radius bounded below and volume going to infinity for which the 1-form Laplacian has spectral gap vanishing exponentially fast in the volume.

We prove that the comparison map from $�G$-fixed points to $�G$-homotopy fixed points, for the $�G$-fold smash power of a bounded below spectrum $�B$, becomes an equivalence after $�p$-completion if $�G$ is a finite $�p$-group and $��H_{\ast }��(�B�;�F_p�)��$ is of finite type. We also prove that the map becomes an equivalence after $��I�\left(�G�\right)�$-completion if $�G$ is any finite group and $��{\pi }_{\ast }��\left(�B�\right)��$ is of finite type, where $��I�$

We prove that semisimple four-dimensional oriented topological field theories lead to stable diffeomorphism invariants and can therefore not distinguish homeomorphic closed oriented smooth four-manifolds and homotopy equivalent simply connected closed oriented smooth four-manifolds. We show that all currently known four-dimensional field theories are semisimple, including unitary field theories, and once-extended field theories which assign algebras or linear categories to 2-manifolds. As an application, we compute the value of a semisimple field theory on a simply connected closed oriented 4-manifold in terms of its Euler characteristic and signature. Moreover, we show that a semisimple four-dimensional field theory is invariant under $��C�P^2�$-stable diffeomorphisms if and only if the Gluck twist acts trivially. This may be interpreted as the absence of fermions amongst the ‘point particles’ of the field theory. Such fermion-free field theories cannot distinguish homotopy equivalent 4-manifolds. Throughout, we illustrate our results with the Crane–Yetter–Kauffman field theory associated to a ribbon fusion category, settling in the negative the question of whether it is sensitive to smooth structure. As a purely algebraic corollary of our results applied to this field theory, we show that a ribbon fusion category contains a fermionic object if and only if its Gauss sums vanish.

We introduce a type of surgery decomposition of Weinstein manifolds that we call simplicial decompositions. The main result of this paper is that the Chekanov–Eliashberg dg-algebra of the attaching spheres of a Weinstein manifold satisfies a descent (cosheaf) property with respect to a simplicial decomposition. Simplicial decompositions generalize the notion of Weinstein connected sum and we show that there is a one-to-one correspondence (up to Weinstein homotopy) between simplicial decompositions and so-called good sectorial covers. As an application, we explicitly compute the Chekanov–Eliashberg dg-algebra of the Legendrian attaching spheres of a plumbing of copies of cotangent bundles of spheres of dimension at least three according to any plumbing quiver. We show by explicit computation that this Chekanov–Eliashberg dg-algebra is quasi-isomorphic to the Ginzburg dg-algebra of the plumbing quiver.